Part of K C Cole’s teaser for tomorrow’s Categorically Not! – Inside Out said:
Sometimes the results are surprising: circles in the plane canâ€™t be turned inside out, but spheres in 3-dimensional space can be.
This is all the license I need to show you* this wonderful 21 minute video showing exactly that (and explaining some rather beautiful mathematics along the way):
Direct link to the Google Video (for larger viewing) here.
Some Related Asymptotia Posts (not exhaustive):